Question

write the coordinates of the vertices after a dilation with a scale factor of 1/4, centered at the origin.

Explanation:

1. First, identify the original coordinates of the vertices:

• Assume the coordinates of the vertices of the rectangle are \(T(- 8,-8)\), \(W(-8,-4)\), \(V(8,-4)\), \(U(8,-8)\).
• The rule for a dilation centered at the origin with a scale - factor \(k\) is \((x,y)\to(kx,ky)\). Here, \(k = \frac{1}{4}\).

Step1: Find the new coordinates of point \(T\)

• For \(T(-8,-8)\), using the dilation rule \((x,y)\to(kx,ky)\) with \(k=\frac{1}{4}\), we have \(x=-8\) and \(y = - 8\).
• \(x_{new}=k\times x=\frac{1}{4}\times(-8)=-2\) and \(y_{new}=k\times y=\frac{1}{4}\times(-8)=-2\). So the new coordinates of \(T\) are \((-2,-2)\).

Step2: Find the new coordinates of point \(W\)

• For \(W(-8,-4)\), with \(x=-8\) and \(y=-4\) and \(k = \frac{1}{4}\).
• \(x_{new}=k\times x=\frac{1}{4}\times(-8)=-2\) and \(y_{new}=k\times y=\frac{1}{4}\times(-4)=-1\). So the new coordinates of \(W\) are \((-2,-1)\).

Step3: Find the new coordinates of point \(V\)

• For \(V(8,-4)\), with \(x = 8\) and \(y=-4\) and \(k=\frac{1}{4}\).
• \(x_{new}=k\times x=\frac{1}{4}\times8 = 2\) and \(y_{new}=k\times y=\frac{1}{4}\times(-4)=-1\). So the new coordinates of \(V\) are \((2,-1)\).

Step4: Find the new coordinates of point \(U\)

• For \(U(8,-8)\), with \(x = 8\) and \(y=-8\) and \(k=\frac{1}{4}\).
• \(x_{new}=k\times x=\frac{1}{4}\times8 = 2\) and \(y_{new}=k\times y=\frac{1}{4}\times(-8)=-2\). So the new coordinates of \(U\) are \((2,-2)\).

Answer:

The new coordinates of the vertices are \(T(-2,-2)\), \(W(-2,-1)\), \(V(2,-1)\), \(U(2,-2)\)